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Theorem csbriotag 6562
Description: Interchange class substitution and restricted description binder. (Contributed by NM, 24-Feb-2013.)
Assertion
Ref Expression
csbriotag  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( iota_ y  e.  B ph )  =  ( iota_ y  e.  B [. A  /  x ]. ph ) )
Distinct variable groups:    y, A    x, B    x, y
Allowed substitution hints:    ph( x, y)    A( x)    B( y)    V( x, y)

Proof of Theorem csbriotag
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3254 . . 3  |-  ( z  =  A  ->  [_ z  /  x ]_ ( iota_ y  e.  B ph )  =  [_ A  /  x ]_ ( iota_ y  e.  B ph ) )
2 dfsbcq2 3164 . . . 4  |-  ( z  =  A  ->  ( [ z  /  x ] ph  <->  [. A  /  x ]. ph ) )
32riotabidv 6551 . . 3  |-  ( z  =  A  ->  ( iota_ y  e.  B [
z  /  x ] ph )  =  ( iota_ y  e.  B [. A  /  x ]. ph )
)
41, 3eqeq12d 2450 . 2  |-  ( z  =  A  ->  ( [_ z  /  x ]_ ( iota_ y  e.  B ph )  =  ( iota_ y  e.  B [
z  /  x ] ph )  <->  [_ A  /  x ]_ ( iota_ y  e.  B ph )  =  ( iota_ y  e.  B [. A  /  x ]. ph )
) )
5 vex 2959 . . 3  |-  z  e. 
_V
6 nfs1v 2182 . . . 4  |-  F/ x [ z  /  x ] ph
7 nfcv 2572 . . . 4  |-  F/_ x B
86, 7nfriota 6559 . . 3  |-  F/_ x
( iota_ y  e.  B [ z  /  x ] ph )
9 sbequ12 1944 . . . 4  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
109riotabidv 6551 . . 3  |-  ( x  =  z  ->  ( iota_ y  e.  B ph )  =  ( iota_ y  e.  B [ z  /  x ] ph ) )
115, 8, 10csbief 3292 . 2  |-  [_ z  /  x ]_ ( iota_ y  e.  B ph )  =  ( iota_ y  e.  B [ z  /  x ] ph )
124, 11vtoclg 3011 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( iota_ y  e.  B ph )  =  ( iota_ y  e.  B [. A  /  x ]. ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652   [wsb 1658    e. wcel 1725   [.wsbc 3161   [_csb 3251   iota_crio 6542
This theorem is referenced by:  cdlemkid3N  31730  cdlemkid4  31731
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-riota 6549
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